This paper is concerned with the persistence of a single species population in a habitat represented by a finite rooted tree with a lethal boundary and whose interior is a Cayley tree. This Cayley tree is a habitat outside which the organisms cannot live. The population is assumed to be governed by an irreducible cooperative system of differential equations that admits zero as an equilibrium solution. The system is determined by five parameters that describes the structure of the tree representing the habitat and the demographic aspects of the population such as migration and reproduction. It is shown that the population is persistent if, and only if, the zero solution is unstable. The stability of the zero solution is determined by the eigenvalue of the Jacobian of the system at zero with largest real part. After obtaining analytically such eigenvalue in terms of the parameters of the system, two criteria for evaluation of persistence are derived. One of them is the minimum patch size, which, here, is defined as the minimum number of levels that makes the tree a habitat capable of sustaining the population.